Fractional Powers of Operators of Tsallis
Ensemble and their Parameter Dierentiation
A. K.Rajagopal NavalResearchLaboratory, WashingtonD.C.20375-5320,USA
Received 07 December, 1998
We develop four identities concerning parameter dierentiation of fractional powers of operators appearing in the Tsallis ensembles of quantum statistical mechanics of nonextensive systems. In the appropriate limit these reduce to the corresponding dierentiation identities of exponential operators of the Gibbs ensembles of extensive systems derived by Wilcox.
I Introduction
Wilcox [1] in 1967 published a seminal paper entitled \Exponential Operators and Parameter Dierentiation
in Quantum Physics". This paper was centered around the following remarkable identity:
If ^H() is an operator depending on a parameter, then
c
@ @
^
Q(;) =, Z
0
duQ^(;) ^Q ,1(
;u) @H^()
@ ^
Q(;u); (1)
where
^
Q(;u) = exp,uH^(): d
From this he went on to obtain several other identities in elegant ways which are all central in the develop-ment of quantum time evolution, Gibbsian ensembles in equilibrium quantum statistical mechanics, pertur-bation expansions, inequalities concerning correlation functions etc., all of which depend on the appearance of the exponential operator of the form introduced in Eq.(1). For a comprehensive account of the rami-cations of this identity, see Appendix A in Vol. I of Grandy's Textbook on Statistical Mechanics [2]. In Ap-pendix D of the Vol.II of his text, he uses this identity to establish inequalities of various covariance functions which are just the quantum mechanical variances and covariances of quantities of interest. It is useful to re-call that the exponential form in the Gibbsian
ensem-ble arises from the principle of maximumvon Neumann entropy,S
1=
,Tr^ln ^; withTr^= 1, subject to the given mean energy of the system, E = Tr^H^, as ex-plained in [2], for example.
In 1988, Tsallis [3] introduced a dierent ensemble for describing a large variety of nonextensive systems for which the exponential form of the operator is re-placed by a monomial fractional power of the form
^ Q
T( ;) =
h
1,(1,q)H^() i
q =(1,q )
the system consists of particles which are either non-interacting or non-interacting with short-range forces. The operator in Eq.(2) replaces the exponential operator in Eq.(1) when statistical expectation values of quantities of interest are to be calculated in the Tsallis formalism. Here the Tsallis entropy S
q =
,Tr(^,^ q)
=(1,q), is maximized subject to the given q-expectation value (a dierent form of the q-expectation value to be de-ned later is sometimes preferred) of the Hamiltonian, E
q = Tr^
q ^
H, besides the usual normalization,Tr^= 1. It is in this form (as well as the dierent form alluded to above) that the formal structure of Statistical Me-chanics is preserved namely - Legendre transform na-ture of the free energy. This ensemble has now been used to develop [4] a Green function theory of many-particle nonextensive systems in much the same way as the Gibbsian ensemble is used in the Green function theory of the corresponding extensive systems. With this development, the program of statistical mechanics of nonextensive systems is achieved in a manner that is formally complete in parallel to the conventional sta-tistical mechanics of extensive systems.
It may be important to give the reader some com-pelling reasons for setting up another ensemble dif-ferent from the traditional Boltzmann-Gibbs ensemble for dealing with nonextensive systems. We rst recall that among the statistical distributions, the exponen-tial - class played very important role in the analysis of many phenomena. (see E. T. Jaynes [5] for a dis-cussion of these aspects). These can all be derived from a maximum entropy principle subject to some constraints in which the entropy functional is chosen to be the Gibbs-von Neumann form, S
1 =
,Tr^ln ^. There are many other probability distributions possess-ing long tails such as Pareto, Levy, etc. which are of the monomial - class, not related to the exponential class. These are not derivable from the maximum en-tropy principle with the Gibbs-von Neumann form for
the entropy functional. These cover many phenomena which do not come under the rubric of \extensive" class of systems which were traditionally treated in physical and other sciences. It is this important gap that the Tsallis ensembles cover. As far as we are aware, there is no mathematicalor physical argument to rule out the applicabilityof the Tsallis ensemble nor do we know any demonstration that the exponential-class covers every conceivable situation in physical and other sciences so that the universality of the Boltzmann{Gibbs ensemble may be considered as the only one paramount form. In discussing sensitivity to initial conditions of nonlinear dynamical systems, similar \exponential"and \power" law sensitivities have been discussed recently [6,7] in the context of the use of von Neumann (Kolmogorov and Sinai) and Tsallis entropies in their quantication. The Tsallis ensemble withq6= 1 deals with Hamiltoni-ans of systems with long-range interactions which may exhibit nontrivialanomalies in their ergodicity and mix-ing properties. For systems which are noninteractmix-ing or interacting systems with short-range forces, one cer-tainly hasq= 1 (Boltzman-Gibbs class).
II Four theorems
In this paper, we rst obtain the counterpart of the Wilcox theorem, Eq.(1), for the Tsallis formof the oper-ator, and then deduce three others. It is not out of place here to mention that some of these have been recently stated without derivation [8] in developing the dynamic linear response of a nonextensive system based on Tsal-lis framework. It is worth pointing out that every one of these theorems has its counterpart in the exponential version, derived by Kubo, Karplus and Schwinger in dierent physical situations of the respective authors' interests. Wilcox's theorem unies all of these in a sin-gle, elegant form, from which all others follow.
c THEOREM I:
@ @
^ Q
T(
;) =,q Z
duQ^
T( ;) ^Q
,1 T (
;u)
@H^~(;u) @
^ Q
(3) where @H^~(;u)
@
=
1,u(1,q) ^H()
,1 @H^()
@
1,u(1,q) ^H()
,1 :
(4) Prof: Let
^ F
T( ;) =
@Q^ T(
;) @
: (5)
Then dierentiating with respect to, interchanging the order of dierentiation, using the denition (4), and using the denition (3) given above, we obtain the dierential equation
@F^ T(
;) @
+q h
1,(1,q) ^H() i
,1 ^ H() ^F
T( ;)
(6)
= ,q
@H^~(;) @
^ Q
T( ;):
Using the explicit form given in Eq.(2), and dening the inverse operator, ^Q ,1 T (
;), in the usual way, we recast Eq.(6) in the form
^ Q
T( ;)
@ @
h ^ Q
,1 T (
;) ^F T(
;) i
=,q
@H^~(;) @
^ Q
T(
;): (7)
From this expression the theorem I is established upon integration of both sides of Eq.(7), because ^Q T(
;= 0) = 1 and ^F
T(
;= 0) = 0.
The above Theorem reduces to the Wilcox theorem [1] when q is set equal to unity.
This derivation is more direct than the one used in Ref.[1,2] for the Gibbsian ensemble. This is our central theorem. From this, we deduce the following theorem by applying Theorem I using a similarity transformation to generate the-dependence of ^H().
THEOREM I I:
If ^Ais an arbitrary operator, then the commutator h
^ A;Q^
T( ;)
i
is given by
h ^ A;Q^
T( )
i =qQ^
T( )
Z 0
duQ^ ,1 T (
u) h
^ H ;A^(u)
i ^ Q
T( u) where
^ Q
T( ) =
h
1,(1,q) ^H i
q =(1,q ) ; ^
A(u) = h
1,u(1,q) ^H i
,1 ^ A h
1,u(1,q) ^H i
,1
: (8)
Proof: We deduce this theorem by taking ^H() in Theorem I as a similarity transformation of the following form ^
H() = (expA^) ^H(exp,A^): (9)
Then it follows that
^
Q(;) = (expA^) ^Q T(
)(exp,A^); (10)
where we used the denition in Eq.(8). Then, we obtain the following expressions that appear in Eq.(3)
@Q^ T(
) @
= expA^ h
^ A;Q^
T( )
i
exp,A^;
and (11)
@H^~()
= expA^ h
^ A(u);H^
i
which in Theorem I lead to the result in Eq.(8). In the limit whenq= 1, this goes to the well-known Kubo identity [2] which was of much use in his theory of irreversible processes.
We now employ Theorem I to deduce another important result concerning the parametric derivative of an Tsallis-expectation value of an arbitrary operator, often useful in computing the linear response function of a Tsallis-mean value of a quantity of interest. We must remark here that this denition of the mean value diers from the denition rst proposed in [3] and has the advantage of having the property that the mean value of a scalar constant, C, is C itself, which was not the case in its original formulation. While this entails some changes in the formalism it does not change the basic Legendre structure of the statistical mechanical principles. For a discussion of the implications and ramications of these aspects, one may refer to [9]. Forq= 1 this goes over to the result for the usual thermodynamic average [2].
THEOREM I I I:
Dene the Tsallis-expectation value of an arbitrary operator ^B [9] as
hBi^ T =
TrB^Q^ T(
;) .
TrQ^ T(
;) (12)
Then
@ D
^ B E
T @
= ,q
Z 0
du nD
^ A(;u) ^B
E T
, D
^ A(;u)
E T
D ^ B E
T o
= ,q
Z 0
du nD
^A(;y)
^B E
T o
; w her e (13)
^A = ^A, D
^ A E
;A(;u) = ^Q ,1 T (
;u)
@H^~(;u) @
^ Q
T( ;u):
Proof: This follows upon dierentiation of the dening expression in Eq.(12) and subsequently using Theorem I directly. Quite often, one uses this expression when ^H() = ^H
0+
V^, and evaluates the expression in Eq.(13) for = 0 to obtain the correlation function of interest.
Finally, we develop a Karplus-Schwinger perturbation type theory for the Tsallis ensemble in the next Theorem. This theorem is not in the form of Theorem I but its proof involves steps similar to the one used in Theorem I and hence its inclusion here.
THEOREM IV: If ^H= ^H
0+ H^
1, then ^ Q T
;H^
0+ H^
1
= ^Q (0) T (
;H^ 0)
,qQ^ (0) T (
;H^ 0)
Z 0
du ^
Q (0),1
R (
;H^)
1,u(1,q)
^ H
0+ H^
1
,1
1,u(1,q)( ^H 0)
^~ H
1( u) ^Q
T( u;H^
0+ H^
1) ; w her e H^~
1( u) =
1,u(1,q) ^H 0
,1
^ H 1
1,u(1,q) ^H 0
,1
: (14)
Here
^ Q
T( ;H^
0+ H^
1) =
1,(1,q)( ^H 0+
H^ 1)
q =(1,q )
; (15)
^ Q
(0) T (
;H^ 0) =
1,(1,q) ^H 0
q =(1,q )
Proof: Consider
@ @
n ^ Q
,1 T (
;H^ 0) ^
Q T(
;H^ 0+
H^ 1)
o
= @
@ n
^ Q
,1 T (
;H^ 0)
o ^ Q
T( ;H^
0+ H^
1) + ^
Q ,1 T (
;H^ 0)
@ @
n ^ Q
T( ;H^
0+ H^
1) o
: (16)
Performing the indicated dierentiations, and after some algebra we nd
@ @
n Q
,1 T (
;H^ 0) ^
Q T(
;H^ 0+
H^ 1)
o
=,qQ^ ,1 T (
;H^ 0)
1,(1,q)( ^H 0+
H^)
,1 ^ H 1
1,(1,q) ^H 0
,1
^ Q
T( ;H^
0+ H^
1) (17)
d Introducing the notation given in Eq.(14), upon inte-gration of both sides of Eq.(17) with respect to, from 0 to , and using ^Q
T(
=;0 ^H) = 1, we obtain the result stated in Theorem IV above.
III Summary
In summary, we have here deduced a set of four theo-rems on parametric dierentiation of the operator den-ing the Tsallis ensemble, which we hope are of use in the same way as the Wilcox theorems were for the operator dening the Gibbsian ensemble.
Several versions of this paper were read by Profes-sor Tsallis. Thanks are due to him for making valuable suggestions to improve the presentation. This work is supported in part by the Oce of Naval Research.
References
[1] R. M. Wilcox, J. Math. Phys.8, 962 (1967).
[2] W.T.Grandy, Jr. Foundations of Statistical Mechanics, Vol. I: Equilibrium Theory, and Vol. II:Nonequilibrium Phenomena, D. Reidel Publishing Co., Boston (1988). [3] C. Tsallis, J. Stat. Phys.52, 479 (1988). Since this
pa-per rst appeared, a large number of papa-pers based on this work have been and continues to be published in a wide variety of topics. A list of this vast literature which is continually being updated and enlarged may be obtained from http://tsallis.cat.cbpf.br/biblio.htm [4] A. K. Rajagopal, R. S. Mendes, and E. K. Lenzi, Phys.
Rev. Lett.80, 3907 (1998). See also an expanded
ver-sion of this work by E. K. Lenzi, R. S. Mendes, and A. K. Rajagopal, Phys. Rev. E59, 1398 (1999).
[5] E. T. Jaynes:Papers on Probabilty, Statistics, and Sta-tistical Physics, edited by R. D. Rosenkrantz, D. Reidel Publishing Co., Boston (1983).
[6] M. L. Lyra and C. Tsallis, Phys. Rev. Lett. 80, 53
(1998).
[7] C. Anteneodo and C. Tsallis, Phys. Rev. Lett.80, 5313
(1998).
[8] A. K. Rajagopal, Phys. Rev. Lett.76, 3469 (1996).
